JEE Main & Advanced Mathematics Three Dimensional Geometry Angle Between Two Lines

Let \[\theta \]  be the angle between two straight lines AB and AC whose direction  cosines are \[{{l}_{1}},\,{{m}_{1}},\,{{n}_{1}}\] and \[{{l}_{2}},\,{{m}_{2}},\,{{n}_{2}}\] respectively, is given by\[\cos \theta ={{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}\].

If direction ratios of two lines \[{{a}_{1}},\,{{b}_{1}},\,{{c}_{1}}\] and \[{{a}_{2}},\,{{b}_{2}},\,{{c}_{2}}\] are given, then angle between two lines is given by     \[\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}.\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\].

Particular results: We have, \[{{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \]

\[=(l_{1}^{2}+m_{1}^{2}+n_{1}^{2})(l_{2}^{2}+m_{2}^{2}+n_{2}^{2})-{{({{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}})}^{2}}\]

\[={{({{l}_{1}}{{m}_{2}}-{{l}_{2}}{{m}_{1}})}^{2}}+{{({{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}})}^{2}}+{{({{n}_{1}}{{l}_{2}}-{{n}_{2}}{{l}_{1}})}^{2}}\]

\[\Rightarrow \] \[\sin \theta =\pm \sqrt{\sum {{({{l}_{1}}{{m}_{2}}-{{l}_{2}}{{m}_{1}})}^{2}}}\], which is known as Lagrange’s identity.

The value of \[\sin \,\theta \] can easily be obtained by,

 \[\sin \theta =\sqrt{{{\left| \begin{matrix} {{l}_{1}} & {{m}_{1}}  \\ {{l}_{2}} & {{m}_{2}}  \\ \end{matrix} \right|}^{2}}+{{\left| \begin{matrix} {{m}_{1}} & {{n}_{1}}  \\ {{n}_{2}} & {{n}_{2}}  \\ \end{matrix} \right|}^{2}}+{{\left| \begin{matrix} {{n}_{1}} & {{l}_{1}}  \\ {{n}_{2}} & {{l}_{2}}  \\ \end{matrix} \right|}^{2}}}\]

If \[{{a}_{1}},\,{{b}_{1}},\,{{c}_{1}}\] and \[{{a}_{2}},\,{{b}_{2}},\,{{c}_{2}}\] are d.r.’s of two given lines, then angle \[\theta \] between them is given by \[\sin \theta =\frac{\sqrt{\sum {{({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})}^{2}}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]

Condition of perpendicularity : If the given lines are perpendicular, then \[\theta =90{}^\circ \] i.e., \[\cos \theta =0\]

\[\Rightarrow \] \[{{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}=0\] or \[{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}=0\]

Condition of parallelism : If the given lines are parallel, then \[\theta ={{0}^{o}}\] i.e.,  \[\sin \,\theta =0\Rightarrow \frac{{{l}_{1}}}{{{l}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{n}_{1}}}{{{n}_{2}}}\].

Similarly, \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\].